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Open mapping theorem (complex analysis)
open-in-new
<h2 id="proof">Proof</h2> <p>Assume f : U → C {\displaystyle f:U\to \mathbb {C} } is a non-constant holomorphic function and U {\displaystyle U} is a <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> of the complex plane. We have to show that every <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> in f ( U ) {\displaystyle f(U)} is an <a href="/facts/Interior_point/5sxGnQfY">interior point</a> of f ( U ) {\displaystyle f(U)} , i.e. that every point in f ( U ) {\displaystyle f(U)} has a neighborhood (open disk) which is also in f ( U ) {\displaystyle f(U)} . </p><p>Consider an arbitrary w 0 {\displaystyle w_{0}} in f ( U ) {\displaystyle f(U)} . Then there exists a point z 0 {\displaystyle z_{0}} in U {\displaystyle U} such that w 0 = f ( z 0 ) {\displaystyle w_{0}=f(z_{0})} . Since U {\displaystyle U} is open, we can find d > 0 {\displaystyle d>0} such that the closed disk B {\displaystyle B} around z 0 {\displaystyle z_{0}} with radius d {\displaystyle d} is fully contained in U {\displaystyle U} . Consider the function g ( z ) = f ( z ) − w 0 {\displaystyle g(z)=f(z)-w_{0}} . Note that z 0 {\displaystyle z_{0}} is a <a href="/facts/Root_of_a_function/mlK3dhoT">root</a> of the function. </p><p>We know that g ( z ) {\displaystyle g(z)} is non-constant and holomorphic. The roots of g {\displaystyle g} are isolated by the <a href="/facts/Identity_theorem/jEu4TXXn">identity theorem</a>, and by further decreasing the radius of the disk B {\displaystyle B} , we can assure that g ( z ) {\displaystyle g(z)} has only a single root in B {\displaystyle B} (although this single root may have multiplicity greater than 1). </p><p>The boundary of B {\displaystyle B} is a circle and hence a <a href="/facts/Compact_set/d0cgXJH7">compact set</a>, on which | g ( z ) | {\displaystyle |g(z)|} is a positive <a href="/facts/Continuous_function/sKbl02pB">continuous function</a>, so the <a href="/facts/Extreme_value_theorem/eSV6P7eh">extreme value theorem</a> guarantees the existence of a positive minimum e {\displaystyle e} , that is, e {\displaystyle e} is the minimum of | g ( z ) | {\displaystyle |g(z)|} for z {\displaystyle z} on the boundary of B {\displaystyle B} and e > 0 {\displaystyle e>0} . </p><p>Denote by D {\displaystyle D} the open disk around w 0 {\displaystyle w_{0}} with <a href="/facts/Radius/9Gr9deYD">radius</a> e {\displaystyle e} . By <a href="/facts/Rouch%C3%A9%27s_theorem/CNXqarau">Rouché's theorem</a>, the function g ( z ) = f ( z ) − w 0 {\displaystyle g(z)=f(z)-w_{0}} will have the same number of roots (counted with multiplicity) in B {\displaystyle B} as h ( z ) := f ( z ) − w 1 {\displaystyle h(z):=f(z)-w_{1}} for any w 1 {\displaystyle w_{1}} in D {\displaystyle D} . This is because h ( z ) = g ( z ) + ( w 0 − w 1 ) {\displaystyle h(z)=g(z)+(w_{0}-w_{1})} , and for z {\displaystyle z} on the boundary of B {\displaystyle B} , | g ( z ) | ≥ e > | w 0 − w 1 | {\displaystyle |g(z)|\geq e>|w_{0}-w_{1}|} . Thus, for every w 1 {\displaystyle w_{1}} in D {\displaystyle D} , there exists at least one z 1 {\displaystyle z_{1}} in B {\displaystyle B} such that f ( z 1 ) = w 1 {\displaystyle f(z_{1})=w_{1}} . This means that the disk D {\displaystyle D} is contained in f ( B ) {\displaystyle f(B)} . </p><p>The image of the ball B {\displaystyle B} , f ( B ) {\displaystyle f(B)} is a subset of the image of U {\displaystyle U} , f ( U ) {\displaystyle f(U)} . Thus w 0 {\displaystyle w_{0}} is an interior point of f ( U ) {\displaystyle f(U)} . Since w 0 {\displaystyle w_{0}} was arbitrary in f ( U ) {\displaystyle f(U)} we know that f ( U ) {\displaystyle f(U)} is open. Since U {\displaystyle U} was arbitrary, the function f {\displaystyle f} is open. </p> <h2 id="applications">Applications</h2> <ul><li><a href="/facts/Maximum_modulus_principle/mVvqMhpP">Maximum modulus principle</a></li> <li><a href="/facts/Rouch%C3%A9%27s_theorem/CNXqarau">Rouché's theorem</a></li> <li><a href="/facts/Schwarz_lemma/70DeQvPJ">Schwarz lemma</a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Open_mapping_theorem_(functional_analysis)/nFChD5j5">Open mapping theorem (functional analysis)</a></li></ul> <ul><li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1966), <i>Real & Complex Analysis</i>, McGraw-Hill, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-054234-1</li></ul>